General manager Ned Colletti has been pursuing Hairston for two years, and the desire increased after Jamey Carroll left the club for Minnesota last month.
Hairston can play all over, which includes shortstop, and Colletti wants veteran protection there with only rookie starter Dee Gordon and rookie backup Justin Sellers on the depth chart.
"Obviously, the L.A. Dodgers are one of the most storied franchises in all of baseball and I'm excited to play there," said Hairston. "It's close to home and we'll have Spring Training in Arizona where I live so I'm happy to stay in the NL West. I have kids in school so it's good to be nearby. But above all, there's a chance to win. You look at the second half they had last year and hopefully we can build on that."
Hairston, 35, split 2011 between Washington and Milwaukee, hitting .270. The Brewers have indicated they also are trying to sign him.
Hairston played all over the field in 2011, but only two games at shortstop. He gives the Dodgers a veteran right-handed bat off the bench. He earned $2 million in 2011.
As for Harang, the Dodgers signed lefty starter Chris Capuano to a backloaded two-year, $10 million deal with a 2014 mutual option last week, and Harang is believed to be seeking a two-year, $12 million deal after going 14-7 with a 3.64 ERA for San Diego, earning $3.5 million.
Harang has been on Colletti's radar since 2009, when he nearly traded for the right-hander at the July 31 Deadline and again during the Winter Meetings.
The club could wind up with Capuano and Harang for less than the $12 million in 2012 salary paid to free agent Hiroki Kuroda in 2011.
Harang, 33, threw 170 2/3 innings this year. Kuroda, who turns 37 next spring, went 13-16 with a 3.07 ERA in 202 innings for the Dodgers.
With Clayton Kershaw, Chad Billingsley and Ted Lilly already in the rotation, the additions of Capuano and Harang would give the Dodgers five starters with at least 170 Major League innings this year, although it would appear to block the path of promising rookie Nathan Eovaldi.
Colletti said the signing of a veteran starter might mean moving Eovaldi to the bullpen.
The Dodgers already have signed five Major League free agents -- Hairston, Capuano, second baseman Mark Ellis, infielder Adam Kennedy and backup catcher Matt Treanor -- and re-signed left fielder Juan Rivera.
Also Monday, the club lost outfielder Jamie Hoffmann when he was claimed off waivers by division-rival Colorado. The Dodgers were attempting to outright Hoffmann to clear a spot on the 40-man roster for one of the potential additions.
Ken Gurnick is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.